Simulation and Theory for Plasmonic Holography I. Background There are three main purposes in the simulation and theoretical aspects of our work in plasmonic holography. They are (1) to guide the experiments and design of devices; (2) to explain experimental results; and (3) to explore the design parameter space that may be outside that explored experimentally. The main goal is to design devices that allow manipulation of the surface plasmon (SP) phase across a “wavefront”, while at the same time enhancing the intensity and/or minimizing losses as the SP wave propagates. Our simulations, using the High Accuracy Scattering and Propagation (HASP) code have explored several areas in this regard: (1) The efficient production of SP waves from gratings; (2) the changes in phase and resulting losses from SP waveguides characterized by having SPs propagate between two metal (in our case gold) films; (3) enhancement of SP intensity by using Fabry-Perot (FP) resonators ;(4) the propagation of SPs in the presence of transverse and longitudinal slits on either top or bottom of the SP waveguide; (5) integration of the above components in a SP modulator device. We describe the results of simulation and theory on these topics in the sections that follow. Our main tool for the required EM simulations is the High-Accuracy-Scattering and Propagation (HASP) code.1 This code was written at the Naval Research Laboratory by one of the co-investigators (MIH), and has been used extensively for investigating advanced photonic materials, especially those that exploit SP processes.2-4 The HASP code is a finite-difference-time-domain (FDTD) code that employs a finite difference algorithm that is similar to the widely used Yee algorithm5, but is much more accurate while still being of first order. The simulations yield the EM fields as a function of the spatial coordinates and time. Frequency content is easily obtained either by FFT techniques (when the incident wave is a short pulse) or by running at a fixed frequency (i.e., the incident wave is highly monochromatic). The code also has specific features to handle the negative-complex dielectric constants that describe metals in the visible and IR regimes where surface plasmons can be supported. This program has been optimized for parallel computation by utilizing Message-Passing-Interface (MPI) programming that has increased the computational speed by an order of magnitude. Typical CPU (per processor) times are now less than 1.0 µsec per spatial point per time step per processor on highly parallelized platforms. Most computations were carried out on the HP 4700 at the Air Force Research Laboratory at Wright-Patterson AFB and on the SGI Altix Ice at the US Army Engineer Research and Development Center (ERDC) of the DOD Supercomputing Resource Center ( DSRC) in Vicksburg, MS, under the DOD High-Performance Computation Modernization Project (HPCMP). Simulations basically amount to “numerical experiments”. It is always challenging to reproduce the experimental conditions in finite-difference simulations. Key difficulties are (1) truncation errors in amplitude and phase brought about by the finite gridding as well as resolution of features on the nanoscale from such gridding; (2) truncating an open system by a finite computational volume, i.e., boundary conditions. The HASP algorithm is a vast improvement over the more commonly used Yee algorithm with regards to

truncation errors, but for the complicated nanostructures simulated we have found a grid spacing of ~.01 λ is necessary to get good resolution. This leads to very CPU and memory intensive computations. The multiple-processor capability of the computers in the DOD High Performance Computation Modernization Project has proven very helpful in this regard. With regard to boundary conditions the main problem is to minimize artificial reflection from the boundaries of the computational volume. These have been minimized by employing the Perfectly Matched Layer (PML) method6 developed over the last 20 years, but still there is some difficulty applying this to a metal with its negative dielectric constant. To circumvent this difficultly we have always surrounded the metallic regions with air, which introduces its own artificial reflection (a couple of percents). Thus some algorithmic development is needed here. Finally, theoretical understanding is very helpful in interpreting experimental and simulation results. A promising approach appears in ref. 7 and related work.8 This theory applies an electromagnetic (em) version of the tight binding (TB) formalism, used extensively in solid-state physics, to the problem of light, and in particular SPs, interacting with sub-wavelength slits and grooves. In this approach the fields inside of defects, such as holes, slits, and grooves, are expanded in terms of the fields produced by a perfect conductor (PC), except with an appropriate boundary condition for the real metal applied at the metal-dielectric interface. Details of this theory appear elsewhere 7,8, and one needs to bear in mind that this theory is most valid when the defects are much smaller than a wavelength. We include theoretical analyses using this approach when applicable, i.e., when there are interactions between slits or grooves with SPs. . II. Launching Surface Plasmons Almost any defect on a metal surface can launch SP’s, but to maximize the intensity for a particular frequency requires a regular periodicity of such defects matched to the SP wavelength. In this report we determine the SP intensity when SP’s are launched by an array of holes, by a grating of slits, and by a single slit. Here one has to trade off the experimental simplicity of employing a single slit with the larger intensities obtainable from the periodic structures that are more complicated to handle experimentally. It is a well-known result from the physics of plasmons that Maxwell’s equations supports surface plasmons9 , i.e., EM fields propagating along but localized near metal surfaces (or metal-dielectric interfaces) (from above and below), where the SP propagation wave number ksp along the surface is given by ksp = (ω/c)(εd εm /(εd + εm)) 1/2 , (1) where ω is the angular frequency, c the velocity of light, εd the dielectric constant of the dielectric (or vacuum), and εm the dielectric constant of the metal. A defect (hole, slit, impurity, etc.) on a metal surface, through interacting with the incident radiation, can launch an SP as long as the condition

kinc,s + kd,s = ksp, (2) is satisfied, where kinc,s is the surface component of the wave number (or photon momentum) of the incident EM field, and kd,s is the wave number (or momentum) transferred to the incident photon by the defect or defects interacting with the photon. Defects on the surface will normally produce a distribution of kd,s, related to the Fourier transform of the defect profile, and the efficiency of coupling to SP’s will be determined by that part of the distribution that satisfies Eq. (2). The most efficient coupling occurs when the defects are uniform and form a regular periodic structure, in which case the kd,s distribution is sharply peaked about kd,s = 2 nπ / ax i + 2 mπ / ay j, (3) where n and m are positive or negative integers, and ax and ay are the periodic lengths in the x and y directions, respectively. Eqs. (1)-(3) infer that one can control the frequency and direction of SP propagation by the direction and frequency of incident radiation and by having a lattice of defects with a specified periodicity. The SP resonance will have a finite width because (1) the dielectric constant of real metals has a small but finite imaginary part in the visible and IR, and (2) the defects, even if perfectly periodic, are not perfect sinusoidals and do not yield delta function kd,s distributions. Furthermore, any departure from pure periodicity will further introduce a finite width to the kd,s distribution. In general Eq. (3) is more of a guide than an exact equation, and to really determine how efficiently certain defects launch SPs requires more accurate theoretical or numerical analysis. We found out rather quickly that simulations for SP generation indicated that periodic slit gratings were much more efficient than arrays of holes. The reasons for this are : (1) more em radiation is incident on a slit of length a then on a corresponding square hole of area a2; (2) all the SP modes produced by slits are directed perpendicular to the slit, whereas holes yield modes in many different directions (see Eq. (3)). In light of this we concentrated our attention on the roles of the input slit widths and periodicity in launching SPs. In optimizing the grating spacing and width of the slits we found that the SP intensity can vary by a factor of ten or more even with relatively small changes in either parameter. Our simulation setup was similar to our original experimental setup: a 200 nm gold (Au) film is atop a glass substrate irradiated by a 780 nm laser from below (i.e., through the glass substrate) at normal incidence. (Fig. 3 in the next sections gives an illustration of this). SPs are launched from the input slits, of width w and periodicity d that go entirely through the Au film. We consider the cases of 4 such input slits and also just a single slit. Surface plasmons will appear on both interfaces of the Au film, but with different wavelengths according to the relation of Eq. (1) (εd = 1.0 for Au-air interface, εd = 2.35 for Au-glass interface). In the near IR |εm| >> |εd| and, by Eq. (1) the SP wave length is close to the wavelength in the dielectric medium (or vacuum), and hence by Eqs. (2)-(3), for normal incidence the hole or slit spacing should be close to wavelength in the

medium. In addition to the slit spacing being related to wavelength, the SP intensity could also depend on the slit width as the shape of the slit could influence the kd,s distribution as well as the spacing. We might conjecture that a slit width of one-half the wavelength should be optimal. Then the spatial profile of the surface might resemble (very roughly) a sine wave of wavelength twice the slit width. This is a very simple argument. Simulations and/or theory should give us a better idea. The optimal spacing and width differs for the two interfaces because the SP wavelength for the same frequency is different. For device applications the “business side” is the one with the Au-air interface, and we might want to suppress the SP’s on the glass side. However, our experimental measurements will involve measuring the interference of the SP’s on both sides of the film, thus here the SP intensity on the glass side is of interest as well.

Figure 1 gives the slit-width dependence, for a single slit, of the Ez field intensity near the Au - glass and Au - air interfaces. The peaks are at about 250 and 400 nm, respectively, for the two interfaces. These are not far off from our conjecture that the optimal width is half the wavelength in the glass or air medium. Note, however, that the maximum intensities there are 20-25% of the incident. This, we will see, is much less than what is obtained with multiple slits. Figure 2 gives the dependence of the field intensities near the Au interfaces on the grating period for a 4slit grating with various slit widths. The expected resonant behavior with period occurs with peaks for the Au-glass interface near 450 nm and for the Au-air

Figure 1: Dependence of the field intensity |Ez|2 on the slit width of a single slit grating near the Au-glass and Au-air interfaces. The incident wave is normal to the surface of vacuum wavelength 780 nm (508 nm in the glass). The field intensities are taken 64 nm from the Au-glass and 48 nm from the Au-Air interfaces.

interface near 750 nm. There is also a resonant behavior on the glass side at about 980 nm spacing, which corresponds to the (2,0) mode of Eq. (3). The resonant intensities are about 0.4 times the incident for Au-glass for mode (1,0) at 450 nm spacing and 0.9 times the incident for the (2,0) mode at 980 nm, and 1.2 times the incident for Au-air at spacing 780 nm, which are several times the maximum single slit result. The optimal width parameters for launching SP’s on the air side are a slit-width of 315 nm and period 780 nm. In a device we would primarily be interested in the SPs only on the Au-air interface. In our experimental setup the SP intensity is measured from an output slit in which case the SPs on both the Au-air and Au-glass interfaces interfere. At the optimal parameters for the air-Au interface this interference is fairly small but could have some effect. In practice a ~ 20 nm Ti or TiO2 layer adhesion layer separates the Au and glass, and this suppresses the SPs on the Au-glass interface even more. III. Surface Plasmon Waveguides

Figure 2: Dependence of the field intensity |Ez|2 on the period, or spacing and the slit width of a 4-slit grating at the same distances from the Au-glass and Au-air interfaces as figure 1. The incident wave is normal to the surface of vacuum wavelength 780 nm (508 nm in the glass).

When one Au film is placed above another a SP waveguide is formed. The wavelength and range is changed from that on a single flat surface. A finite length of such a waveguide can thus be used to change the phase of a SP propagating in the waveguide as opposed to what it would be without the top film or with respect to another parallel waveguide with a different gap. We thus have a means of manipulating the interference emerging from a number of such waveguides, which is one main goal of our plasmonic holography effort. However, we must know in advance how the phase changes with waveguide geometry and also assess what losses may be suffered. The material of the top film may alternatively be a dielectric or a metal. In the case of a Au bottom film, we found that a Au top film is most advantageous. In fact, dielectric top films cause the SPs to be scattered away from the surface. Therefore we will concentrate our discussion on Au-air-Au SP waveguides. Our ultimate SP Modulator, in fact, employs a flat bottom Au film and a top film consisting of various Au strips parallel in the direction of SP propagation, i.e., a parallel series of SP waveguides. We now, however, consider the properties of such waveguides infinitely extended in the lateral direction to understand their properties as a function of the air gap in between. Our waveguide setup, including the input slits and substrate, appears schematically in Fig. 3.

We now assess how the gap and length of the top film affects the intensity and phase of the emerging SP. We first calculate the real and imaginary parts of the wave vector, and hence the wavelength, for SPs propagating between two infinitely extended, infinitely

Figure 3: Schematic SP waveguide configuration. SPs are launched from the light incident on the input slits etched on the gold film atop a glass substrate. As the SP propagates through the wave guide its wavelength and intensity are altered.

thick Au films as a function of the air gap. This can be done analytically with the results shown in Fig. 4 for incident light of vacuum wavelength 780 nm.

Fig. 4 shows that the SP wavelength and range decrease as the gap narrows. The decreasing wavelength is desirable in that it will yield a more rapid change in phase (relative to a SP on the flat surface) per unit length of waveguide. On the other hand losses are more severe for narrow gaps. The phase change emerging from such a waveguide is proportional to the length and the difference between the waveguide and without-waveguide wavelengths (the latter is about 762 nm). In terms of efficiency one figure of merit might be the phase change for, say, 50% losses. According to the results of Fig. 1 it turns out that gaps between 50 and 200 nm are the most efficient. Of course the thicknesses of the Au films and the length of the waveguide have some influence on the phase change and losses, but the main influence is essentially the dispersion relation indicated by Fig. 4.

Fig. 4: SP wavelength and range as a function of the gap between infinite gold layers for 780 nm incident light.

The solid curves in Figure 5 illustrate the transmission and phase change predicted by FDTD simulations for the setup of Fig. 3 with a 200 nm lower Au film and a 4 µm long thick (~3 µm) upper film. The dotted curves give the transmission and phase change

when the thick upper film is replaced by a 200 nm film (which is what is used in our experiments). For the thick upper film the transmission generally increases with gap size and the phase shift decreases, as expected from Fig. 4. For the thin upper film there are large fluctuations from this trend, especially with respect to transmission, e.g., the peaks at 120 and 600 nm gaps. The difference between these sets of results is due to interference between the SPs propagating on the top of the top film with that propagating on the lower film. This follows from an examination of the transmission (and phase) as a function of the length of the wave guide. Figure 6 gives the transmission and phase as a function of waveguide length when the upper film is 200 nm thick and the gap is 120 nm. In earlier simulations with the thick upper film we found that to a good degree the phase linearly increased with length whereas transmission was approximately constant to within 20%. Now with the thin upper film the transmission is almost zero when the length is about 1700 nm with maxima near zero and 3400 nm. The maxima and minimum is the signature for

Fig. 5: SP transmission and phase shift after traversing 4 µm of the waveguide of Fig. 3 where the bottom film has a thickness of 200 nm and the top a thickness of 3 µm (solid lines) or 200 nm (dashed lines).

interference between the SP waves on the top film and the bottom film. There is also almost a 360 degree jump in phase at the transmission minimum. The vertical e-folding distance for SPs of this frequency on a Au film is about 600 nm. Thus when the gap distance plus 200 nm upper film thickness is less than or comparable to this the SPs on the upper and lower films can interfere very strongly. This would not happen with the thick upper film because the incident SP would have very little amplitude at the vertical displacement of the upper film. Given the dispersion results of Fig. 4 the difference in wavelength between SPs propagating in the gap and on a flat Au surface would imply a maximum-minimum separation of 2087 nm. The corresponding value from the simulations is 1600-1800 nm. The finite length of the waveguide and reflection and scattering at the ends may help account for the difference. To fabricate devices with easily controllable phase and transmission properties we may wish to suppress the large fluctuations in Fig. 6. One way to do this is to insert a dielectric region between the top film and the upper vacuum (or air). A 20 nm TiO2 layer was previously used as an adhesion layer between the bottom gold film and glass substrate (see Fig. 3). This layer also effectively suppressed SP waves on the glass-air interface. The following table, from analytic calculations, gives the range of SPs propagating on an infinite Au film covered by TiO2 layers of various heights.

Figure 6: The transmission and phase shift for a SP waveguide with a 200 nm upper Au film and a gap of 120 nm as a function of length of the waveguide.

Table 1: Range of surface plasmons (vacuum wavelength 780 nm) for an infinite gold layer covered by various heights of TiO2. Height of TiO2 Layer (nm) Range of Surface Plasmons (µm) 0 42.8 5 31.9 10 23.6 20 12.5 30 6.1 40 2.9 50 1.6 Table 1 shows that the range of SPs can be seriously curtailed by a TiO2 layer. The effectiveness of this layer in eliminating the interference effects in Figs. 5 and 6 depends on the length of the SP waveguide. For example, if the waveguide has a length of 4 µm, as in our simulations, one would need a TiO2 layer of > 35 µm. A 20 nm TiO2 layer, however, would be sufficient for a 20 µm waveguide. IV. Fabry-Perot Enhancement While large gaps do not change the phase (much), they can function as Fabry-Perot (FP) resonators and greatly enhance the SP intensity if top Au films are placed, say, directly above the input slits. The FP effect predicts that gap widths of ~ nλ/2 should be optimal. Fig. 7 illustrates the power converted to SPs from our FDTD simulations. These simulations confirm that the SP intensity was enhanced by a factor of 2.5 for the optimal gap widths, which occur at a periodicity λ/2. A similar enhancement should also occur at the output slits making for an overall enhancement of a factor of 6. Experimental

Fig. 7 : The relative SP power from FDTD simulation, as determined by the summed Poynting vector in the z direction from 0 to 600 nm above the Au surface, as a function of the gap between the Au surface and a top Au film of 150 nm thickness. The top film is placed directly above 6 input slits of width 380 nm and periodicity 830 nm and the plotted power is that relative to having no top film. The slits are illuminated from below as in Fig. 3.

measurements are more favorable indicating an enhancement factor of 15. Details of our FP study appear in ref. 10

One should note that the FP effect does not directly enhance SP production. It enhances the radiant fields in the gap region that interact with the input slits, which in turn enhances SP production. If there is a FP resonator above output slits a reverse enhancement occurs: The SP interacts with the slits producing a radiant field that is enhanced by the FP effect, which in turn is transmitted through the output slits to be detected. The results in Fig. 7 represent the direct SP production step but not the reconversion at the output slit that results in our experimental detection. V. Role of Transverse and Longitudinal Slits In the remainder of our theoretical and numerical analysis we will consider components of a Surface Plasmon Phase Modulator (SPPM), which we have also fabricated experimentally. A schematic diagram appears as Fig. 8.

The main purpose of the alternate straight and curved beams, which are types of longitudinal slits and grooves in the top film, is to create phase differences between alternating sections of the wavefront in the SPPM. This leads to a controllable diffraction pattern – a main goal of our plasmonic holography. Transverse slits are used for the incoupler of Fig. 8, for out couplers to measure the SP radiation, and could also used to separate sections of devices like the SPPM. Later in this report we will describe FDTD simulations of the SPPM. In this section we mainly consider the effect of transverse and longitudinal slits and grooves on the propagation of SPs. First we consider the losses produced by a single slit.

Fig. 8 : Schematic top view of Surface Plasmon Phase Modulator. The top gold film is 200 nm thick with alternating straight and curved beams that are 1000 nm wide and 20 µm long with 200 nm air gaps in between. The gap between the bottom and top films is 250 nm, and the SiO2 lens has a radius of 8 µm and is on the same level as the air gap, and the curved beams dip 100 nm .

Figure 9 shows the loss of SP (vacuum wavelength 732 nm) intensity, as predicted by FDTD simulation, as it transverses a single slit of various widths on a flat Au film. The SP incident on the slit is generated by an appropriate current source placed at z = 0 in the plot. The intensity, as measured by the longitudinal Poynting vector, Sz averaged over 600 nm (roughly the vertical decay length of the SP field in the vacuum) above the gold film, normalized to the Poynting vector 3.1 µm from the current source when there is no gap. The current-launched SP propagates on the Au film over a single transverse gap, of various widths, where the beginning of the gap is positioned 13.6 µm from the current source, as shown on the diagram at the top of the figure. Note that losses exist even when the gap is absent, with the SP having a range of about 34 µm at this frequency. Losses occur here because the dielectric constant of Au at this frequency has an imaginary part,.i.e., absorption.

Fig. 9: Surface plasmon intensity, for various transverse gap widths, as measured by the averaged Poynting vector component Sz from zero to 600 nm above the Au surface, as a function of distance from the current source and gap width (w). The relative value of Sz is normalized to a value of 1.0 for the averaged Poynting vector 3.1 µm from the current source when there is no gap. The diagram at the top illustrates the gold film (filled in blue) relative to the z coordinate on the abscissa and the position of the gap at 13.6 µm from the current source. The Au film is 200 nm thick. The numbers labeling the curves are the widths of the gap in nanometers.

Figure 10 gives the SP transmission across the gap, as a function of gap width. The losses shown in Fig. 10 are those we attribute just to the gap itself, which includes the effects of reflection of SPPs and scattering of electromagnetic energy into the vacuum. We extract the transmission across the gap by comparing the Poynting vector Sz, averaged as in Fig. 9, at a point 2.9 µm after the beginning of the transverse gap (corresponding to the 16600 nm mark in Fig. 9) with that of the zero-gap result at the same point, which is thus normalized to unity. While the Poynting vector at this “observation point” contains the effects of further losses just from the propagation on the gold surface from the end of the gap to this observation point, the renormalization removes this effect that is not related to the gap itself. The simulations use a spatial mesh of (6 nm)3 and a time step of 1/200 of the wave period. When compared with results of a (10 nm)3 and time step of 1/120 of a period, the transmission increases by 3-5%. Thus we estimate the numerical error in the transmission as 3-5%. Fig. 10 also gives our corresponding experimental measurement of losses due to the gap. The setups of the simulation and experiment differ slightly – e.g.,

Figure 10 : Surface plasmon transmission across a transverse gap as a function of the gap width and our experimental results. The transmission is relative value of Sz, averaged vertically as described in Fig. 9, but normalized to a value of 1.0 for the averaged Poynting vector 16.6 µm from the current source when there is no gap. Results are also shown using the “tight-binding”-like theory of ref. 7 for the total SPP transmission, losses due to scattering of the electromagnetic energy from the SPPs into the vacuum at the gap, and the back-reflection of SPPs from the gap.

the gap in the simulation is 13.6 µm past the current source, whereas it is about 18 µm past the end of the grating in the experiment. Also the experiment measures intensity by measuring the light coming from the output slit (30 µm past the grating) whereas the simulation directly measures the SP intensity by calculating the Poynting vector. To make a meaningful comparison we normalize the experimental intensities to the simulation result for the zero-width gap. This isolates the effect of the transmission across the gap and eliminates the effects of further losses because of the propagation distance between the gap and the output slit and of the output slit itself (assuming that the output light intensity is proportional to the SP intensity). The simulation results in Figs. 9 and 10 exhibit two trends. As might be expected from scattering or diffraction arguments, initially the transmission rapidly decreases with gap size. However, as the gap approaches λSP/2 (356 nm) the decrease quenches somewhat. From gap widths λSP/2 to λSP the trend repeats: a decrease in SP intensity with w followed by a leveling off as the gap approaches λSP. In fact, there is actually a local maximum near λSP. Overall there is a general increase in losses with increasing w, but superimposed on this is a reflective interference effect of period λSP/2 similar to that of a Fabry-Perot (FP) resonator. The experimental measurements, while generally indicating more transmission than the simulations, exhibit a similar alternating minima – maxima pattern. The minima are close to the minima in the simulations, but the maxima appear closer together than the simulations - at about 450 nm and 620 nm gaps, as opposed to about 355 nm (here actually a leveling in the decrease) and 720 nm in the simulations. The general decrease of transmission with gap along with local maxima at ~nλSP/2 is argued qualitatively as follows. The SP field has a longitudinal electric field and the finite conductivity at the relevant frequency yields a current flow. The coupling between two Au sides would then be capacitive (via Coulomb fields) and we would expect strong capacitive coupling for tiny gaps (typically λ/8 imperfections do little damage to the wave) and weakening for larger gaps. The SP field also has transverse E and H components, and this part resembles an ordinary propagating plane wave in which case the gap would act like a Fabry-Perot (FP) resonator giving an alternating maximum – minimum structure with gap width. F. López-Tejeira, F.J. García-Vidal, and L. Martín-Moreno7 (which we refer to as LGM) study the interaction of SPs with slits or grooves using an analogy of the tight-binding method used in solid-state physics. (We will refer to this theory by the shorthand TB). Here the fields in the grooves are expanded in modes similar to those enclosed in perfect conductors, and the fields are matched at the top and bottom boundaries of the metal to calculate the fields in all space. The dotted curves in Fig. 10 give the results of this approach for SPs propagating on top of a 200 nm gold film traversing a single gap of various widths. The losses in this approach due to reflection and scattering to the vacuum also appear in the figure. While there are some differences between the TB results and simulations (e.g., the simulations show less transmission for small gaps) the trends are in good agreement, especially for the muted FP effect mentioned above. In the usual FP effect minimum reflection and maximum transmission occurs at half wavelengths. In our situation the reflection is evidently always small (Fig. 9), and losses are almost entirely

due to scattering of the SPs into the upper vacuum, which is minimized at gaps of multiples of half wavelengths. The simulations, theory, and experiments all resemble the usual FP case in that transmission minima occur near odd multiples of quarter wavelengths. The properties of series of slits or grooves are also of interest. Series of transverse slits or grooves are used for the input slits of the SPPM (Fig. 7). More input slits enhance the SP intensity simply because they take in more incident radiation. But this is limited, as we shall see, by the reduced range of a multi-slit array, which comes about from the scattering of SPs into the vacuum and back reflection. Longitudinal grooves/slits on the same device on the upper film (e.g., the “beams”) are used for phase manipulation, and again a reduced SP range is a limiting factor. We first consider series of transverse slits. Just as the wavelength and range in SPs is altered by a SP waveguide, they are also altered in a region of periodic slits. These can be calculated by the LGM method. Doing this we immediately find that ranges are markedly reduced but the wavelengths only slightly changed. Therefore they are not competitive with the waveguides (of gaps <250 nm) for changing phases. However slit or groove arrays can be optimized as in-couplers. S.T. Koev, A. Agrawal, H.J. Lezec, and V. Aksyuk11 (referred to as KALA) suggest that the incoupling efficiency is a sensitive function of the range of the SPs in the grooved region, with the optimal coupling occurring when the range associated with losses from scattering at the grooves is equal to the range associated with losses on a corresponding flat surface. They also measure the ranges and incoupling efficiencies for various groove widths and heights. The results of the range measurements11 and LGM predictions are shown in Fig. 11. The agreement is rather good except for the theory predicting longer ranges than experiment on the large groove height end. This encourages the use of the rather computationally simple LGM theory in assessing the effects of groove geometry.

In Fig. 11 the groove spacing is 560 nm, at which periodicity, as stated in ref. 11, diffraction to higher orders is eliminated. The SP range at this frequency on a flat Au surface is 43.0 µm, meaning that the ideal range for incoupling in Fig. 10 occurs when the range is 21.5 µm because there the ranges produced by the scattering of the grooves and that on a flat surface are equal. The calculations and theory are for illumination on the topside of the grooves, which is opposite that illustrated in Fig. 3, but used in the latest versions of the modulator. The heart of the SPPM (Fig. 8) are the longitudinal beams that vary the phase across the wave front. So we now consider the effect of longitudinal grooves and slits on SP propagation. Fig. 12 shows the SP intensity as it propagates along a series of longitudinal gold strips separated by air gaps as shown in the diagram at the top of the figure. In our experimental arrangement we usually employ about 8 or 9 strips of width 0.75 µm (as opposed to three shown in the figure). The simulations employ periodic boundary conditions in the x direction, which corresponds to an infinite series of such strips (and gaps). The SPs in the simulation are current-launched, and the spatial grid used is (10 nm)3 and the time step 1/120 of a period. By comparison to simulations with a (15 nm)3

grid , we estimate that the error in the ranges in the inset of Fig. 12 at 10-15%. The strip structure starts 3.6 µm from the end of the launching current region and extends 11 µm, as in the experiment, at which point an ungrooved Au film resumes. Extending the grooved film further to the end of the 30 µm box has negligible effect on the curves of Fig. 12. To a good degree all the curves in Fig. 12 have exponential decay, even when there is no gap, as previously discussed. The inset of Fig. 12 gives the range, as a function of gap size, based on a least squares fit, for the five gap sizes considered. The losses here may be described in terms of scattering from the strip structure and modifications of the SP propagation vector due to the periodic strip structure, as illustrated by the agreement with the theoretical curve (labeled “TB”) in the inset, as will be described below. The inset to Fig. 12 also shows results of experimental measurements for a 9 strip SP waveguide. The experimental range is extracted from the transmission data at the output slit taking into account that the strip region is 11 µm long. As with Fig. 3 the experimental measurements have been normalized to simulation results for the zero gap case, and thus the ranges in the inset correctly shows the difference in ranges between the simulations and experiment (and also difference between the finite gap cases and zero gap).

Fig. 11: The e-folding range of SPs, with frequency corresponding to 780 nm vacuum wavelength, propagating in a region of grooves spaced at 560 nm, as calculated by the TB theory (LGM) of ref. 7 and measured in ref. 11, as functions of groove width and height. For reference the range for these SPs on a flat Au surface is 43.0 nm. The grooved region used in the theory and experiment is 40 µm long (72 grooves).

Generally there is good agreement between experiment, theory, and simulation for the overall decrease of range with gap size. Simulations give a fairly uniform decrease in range with gap width, while the theory and experiment indicate some maximum – minimum structure (inset, Fig. 12). However the positions of the maxima and minima in theory and experiment differ. For device applications we would like ranges > 15 µm, which indicates that gaps up to about 150 nm could be tolerated. One reason for the disagreement between simulations and experiment could be due to the simulation employing an infinite number of strips while the experiment employs a finite number (8 or 9). The finite number of strips used in the experiment would necessarily lead to a somewhat shorter SP range than in the simulation. The simulations also indicate a range of about 28 µm for the ungrooved case compared to a range of 33 µm gotten from the imaginary part of the SP wave number at 732 nm for a flat Au film.

Fig. 12. Surface plasmon intensity, for current launched SPs, as measured by the summed Poynting vector component Sz from zero to 600 nm above the Au surface, as a function of distance from the beginning of a striped region and gap width. The number associated with each curve gives the gap width in nm, whereas the Au strip width is always 0.75 µm. The position of the gap is illustrated at the top of the figure with the filled region indicating the Au surface. The inset gives the e-folding range as a function of gap width (see text) from simulations and the theory of LGM – labeled “TB theory”. Error bars of ± 3% (too small to be seen) are assigned to the experimental points in the inset.  

Ebbesen et al. 12 have discussed the usefulness of metal strips as SP waveguides. Previous treatments13 of SP propagation on strips have mainly concentrated on a single metal strip, whereas our case employs multiple (e.g., 8) strips. The “tight binding” theory of LGM does not directly address longitudinal gaps but restricts itself to SPs normally incident on transverse gaps. We extend this approach as follows: We use the methods of ref. 7 to calculate the transmission and reflection of SPs normally incident on a series of n transverse gaps (usually n = 8) with the same width and spacing as the longitudinal gaps we are considering. We then approximate the problem in a 2D way where an electromagnetic (em) wave of wave number kSP and frequency ω in the metallic regions encounters a dielectric region with a complex dielectric constant and width ( n- 1) d + a , where a is the gap width of the slits and d is the periodicity of the slits. The effective dielectric constant in the metallic regions of width d – a is determined by kSP and ω, and we calculate what effective complex dielectric constant in the region of width ( n- 1) d + a would account for the transmission and reflection results of the TB theory. (This has a significant imaginary part since most of the losses are due to scattering, not reflection, and this is treated as an effective absorption). We then estimate a typical change in the SP propagation vector due to interaction with the series of slits represented by a corresponding series of alternating dielectric regions, which is in the x direction, i.e., normal to the slits. These alternating regions consist of a metallic region d – a = 0.75µm wide followed by a region of width a region ( n- 1) d + a of its dielectric constant. The change in the wave number can have both real and imaginary parts where the latter will correspond to the attenuation the “diverted” SP suffers as it propagates in the direction perpendicular to the slits (as well as parallel). One way to estimate the change in the SP propagation vector ΔkSP , appropriate for a periodic collection of alternating (complex) dielectric regions, is to solve for Bloch states. In this case one obtains both real and imaginary parts of the Bloch q vector (which here only has an x component). The real part of q, which we look for in the first Brillouin zone, indicates the typical displacement of the wave vector in the x direction, and the imaginary part gives the decay constant of the SP wave amplitude in the x direction. The overall decay constant κ for the SP intensity I as measured along the principal propagation direction along the z axis is then given by

 where the factor two appears because of converting decay constants for the amplitude to that for the intensity. The e-folding range is simply 1/κ. With this ansatz the Re(q) ranges from .00038 nm-1 for 50 nm air gaps to .0026 nm-1 for 400 nm gaps, and Im(q) ranges from .000055 nm-1 to .00029 nm-1- over the same interval of gap widths, compared to kSP = 0.00882 + 0.0000147 i nm-1. The ranges obtained from Eq. (4) appear in curve labeled “TB theory” in the inset of Fig. 12. The theoretical results are in good agreement at the experimental points (which have about ± 3% error bars) and with results obtained from simulations, except the simulation results appear to be slightly too low for the narrowest gaps. Again the LGM approach gives accurate results with little computational effort. VI. Surface Plasmon Phase Modulator Simulations

We now consider the phase modulation and losses from the SPPM of Fig. 8. From the previous results we have an idea on the performance of the various components, so now we have a chance to assess the modulator as a whole. We approach this assessment in two phases. The first considers the role of the alternate straight and curved beams essentially in isolation, while the second looks at the performance of the full SPPM. We first consider the SPPM of Fig. 8, but without the focusing SiO2 lens, and with the alternate beams and the gold films extended indefinitely in the lateral (i.e., x) direction. We also use current-launched SPs to eliminate the input slits and to isolate the effect of SPs impinging on the region of the beams. The beams in Fig. 8 are 1.0 µm wide with 200 nm air spaces in between. The gap between the bottom film and the straight beams is 250 nm (as in the fabricated devices) and the top beams are 200 nm thick and 20 µm long. The curved beams are configured like the straight beams except they have a parabolic shape lengthwise with a dip of 100 nm midway. The simulations have a computational volume that extends 2400 nm laterally (a unit cell in the x direction consisting of one straight beam, one curved beam, and two air gaps), 72 µm in the z direction, with the Au film ending at 64 µm, and 3.0 µm vertically (y direction). Perfectly Matched Layer (PML) boundary conditions6 at the y and z extremities minimize artificial reflections from the computational box. Fig. 13 gives the SP intensity, as determined by the total Sz summed up to 250 nm above the bottom Au film over the full lateral extent of the film, as a function of the z distance from the beginning of the beam region. This is normalized to 1.0 at the beginning of this region. The overall effective SP range in the beam region is 9.1 µm, similar to the results in Fig. 12. The main purpose of the alternate straight and curved beams in the SPPM is to create a phase difference across the SP wavefront. Fig. 14 gives the phase angle of Hx across the unit cell of one curved beam and one straight beam and two separating air gaps at various longitudinal distances (referred to the beginning of the beam region). The phase profiles shown are at 8 µm, i.e., near the middle of the beams, at 28 µm, i.e., 8 µm past the end of the beams, and at 48 µm, well past the beam region. The maximum phase difference between the beams is about 220 degrees at 8 µm, 190 degrees at 28 µm, and 150 degrees at 48 µm. As might be expected the transition in phase is sharpest in the beam region, and the phase difference decreases and the transition smoothens as one proceeds further past the end of the beams. This “smearing” of phase differences is likely due to the diffraction and scattering produced by the beams. These simulation results demonstrate that the SPPM design of Fig. 8 is capable of producing phase difference of the order of π radians across distances of the wavefront of order of the wavelength. The principal effect of the phase differences across the wave front should be some sort of diffraction pattern. Fig. 15 gives a 3D plot of the SP intensity, as determined by the Poynting vector in the direction of SP propagation, Sz, summed over 250 nm above the bottom Au film, as a function of x and z.

Fig. 13 : The overall SP intensity as it propagates along the beam region (and beyond) of the SPPM of Fig. 3 as modified by the description in the text. The intensity is normalized to that at the entrance to the beam region.

Fig. 14 : The phase of Hx 60 nm above the bottom Au surface laterally across the unit cell (x direction) for three different distances from the beginning of the beam region of Fig. 8. The numbers labeling the curves indicate the distance in µm from the beginning of the beam region. The beam region extends 20 µm. Laterally the regions corresponding to the curved beam, the straight beam, and the air gaps in between are indicated.

In Fig. 15 z coordinates of between 12000 and 32000 nm define the beam region and at z > 32000 nm the SPs are propagating on a bare Au film. The distinct maxima that appear for z > 32000 show the near- and intermediate-field diffraction pattern. This result indicates that the alternating curved and straight beam design used in the SPPM has the capability of producing a controlled diffraction pattern (controlled by the size, shape, and spacing of the beams), which effectively in the far-field controls the direction of the SP “jet”. The fabricated SPPM differs from that employed in Figs. 13-15 in that there are a finite number of beams (actually eight in our actual device) and the presence of an SiO2 lens just after the beam region. (See Fig. 8). In addition to the SiO2 lens, there are also SiO2 posts of 2 µm lengths supporting the ends of the beams. Simulations without the beams

Fig. 15 : Three dimensional plot of the x and z dependence of the SP intensity as measured by Sz summed over 250 nm above the bottom Au film. The x and z axes labels are in nm, and the intensity is in arbitrary units. The beam region is z = 12000 nm to z = 32000 nm. The curved beams extend from x = 0 nm to x = 1000 nm, and the straight beams from x = 1200 nm to x = 2200 nm.

x

z

or the posts but just with the SiO2 lens (of radius 8 µm) indicate a focal spot centered about 25 µm past the beginning of the lens. This is close to the focal point one would predict from a lens of this shape when the index of refraction changes by that indicated from the change of the real part of the wave number for SPs propagating along a Au-SiO2 interface to a Au-air interface (i.e., nAu-SiO2 / nAu-air = 1.54), which would be ~30 µm. This latter number is really the small incident angle limit and since the lens is cylindrical there is some aberration that would smear out and reduce the distance of the center of the focal spot somewhat. What is really of interest is how the SPPM varies the phase across the wavefront and losses under the beams. Fig. 16 plots the relative phase and intensity profiles across the beam front in the beam region at 6 and 18 µm from the beginning of the beams for the SPPM of Fig. 8.

Fig. 16 : The relative phase (solid lines) and intensity profiles (dashed lines) of Hx 50 nm above the bottom Au surface across the wavefront in the beam region for the SPPM of Fig. 8. The phase difference is referenced to the phase at the center of the central straight beam (at 12000 nm in this figure) for the given distance from the beginning of the beams, and the intensity (in percents) is relative to that at the center of the central straight beam 2µm from the beginning of the beam region. A half profile is shown as the Hx field is symmetric about the longitudinal center line (x = 12000 nm in the figure). The positions of the curved and straight beams are indicated and the distances on the abscissa indicate the distance from the edge of the actual SPPM.

The maximum phase difference between a curved and straight beam segment of the wavefront is about 40 degrees at 6 µm into the beam region and about 70 degrees at 18 µm (i.e., 2 µm from the end). These are not as large phase differences as for the idealized infinitely periodic beams of Figs. 13-15, but still a significant phase difference. The losses are similar. The intensities are significantly larger under the curved beams. This was also true for the periodic beams as one can see in Fig. 15 that there is more intensity in the beam region (z = 12000 to 32000 nm) under the curved beam (x = 0 to 1000 nm) than under the straight beam (x = 1200 to 2200 nm). Unfortunately this hampers the performance somewhat as the phase difference would play the largest role in diffracting the SP jet when the intensities are roughly equal.

Fig. 17 : 3D simulation of the SP intensity from the modulator of Fig. 8. The beam region and SiO2 lens region are outlined by dotted lines. The region in the dotted circle is explained in the text. Only half the width is shown because the intensities are symmetric about the center line at 12000 nm in the figure.

Fig. 17 gives a 3D plot of the SP intensity for the SPPM as measured by the total Sz summed up to 500 nm above the bottom Au surface. Although more subtle than in Fig. 13, multiple peaks are seen past the modulator and lens (note circled region). The intensity distribution after the lens may explain the multiple peaks of out-coupled light seen in the experimental data and the combination of effects coming from the phase changes in the modulator (i.e., beam region) and focusing from the lens. The fact that the contrast in intensity here is less than for the idealized structure (Fig. 15) is a consequence of the smaller phase differences between curved and straight beam sections of the wavefront (compare Figs. 14 and 16). VII. Concluding Remarks on Simulation and Theory Our simulation and theoretical investigations of the SPPM and related components has helped determine the acceptable parameter space for our device as well as plasmonic devices in the future that may employ similar components. Our main goal has been to manipulate the spatial variation of the SP phase while maximizing SP intensity. We have determined the optimal way to launch SPs using multiple input slits. The optimal spacing (for normal incidence) is equal to the SP wavelength with slit widths roughly half the SP wavelength. Placing a second Au film as a Fabry-Perot resonator can enhance the SP intensity by an order of magnitude. The optimal air gap in the SP waveguide is 50-200 nm as gaps smaller will lead to unacceptable losses, while larger gaps will not yield much of a phase shift. Thin upper films hamper the control of the phase (and intensity) because of interference of SPs on the upper film, but this can be mollified by coating the upper film with a dielectric such as TiO2. We can significantly vary the phase across the wavefront by employing, e.g., alternating straight and curved “beams” to effectively vary the gap size of neighboring “waveguides” across the wavefront. Our study of the behavior of SPs propagating on longitudinal strips has shown that we should not have gaps much larger than 200 nm between such beams or else losses may be unacceptable. Finally our investigation of both idealized and realistic SPPMs has confirmed that phase differences of 90 degrees or more are obtainable between pairs of such beams, but a disparity in intensity under the beams may hamper the modulators performance. If, as likely, the disparity comes from different interference with SPs atop the upper film, the cure may be to coat the upper film with something like TiO2 to suppress the upper SPs. Also a wider modulator may help as it would more resemble the idealized version assessed in Figs. 13-15 as these give larger phase differences. References 1. B.S. Dennis, V. Aksyuk, M.I. Haftel, S.T. Koev, G. Blumberg, Journ. Appl. Phys. 110, 066102 (2011); M.I. Haftel, C. Schlockermann, and G. Blumberg, Phys. Rev. B74, 235405 (2006). 2. M.I. Haftel, C. Schlockermann, and G. Blumberg, Applied Physics Letters 88, 193104 (2006). 3. Shannon M. Orbons, Ann Roberts, and David N. Jamieson, Michael I. Haftel, Carl Schlockermann, Darren Freeman and Barry Luther-Davies, Applied Physics Letters

90, 251107 (2007). 4. S. M. Orbons, M. I. Haftel, C. Schlockermann, D. Freeman, M. Milicevic, T. J. Davis, B. Luther-Davies, D. N. Jamieson, and A. Roberts, Optics Letters 33, 821 (2008). 5. K. Yee, IEEE Trans. Ant. and Prop. 14, 302 (1966). 6. J.-P. Berenger, J. Comp. Phys.,110, 185 ( 1994); S.D. Gedney, IEEE Trans. Ant. and Prop. 44, 1630 (1996). We use the latter version of PML. 7. F. López-Tejeira, F.J. García-Vidal, and L. Martín-Moreno, Appl. Phys. A89, 251 (2007). 8. F. Lopez-Tejeira, F.J. Garcıa-Vidal, L. Martın-Moreno, Phys. Rev. B 72, 161 405(R) (2005). 9. H. Raether, Surface Plasmons (Springer, Berlin, 1988). 10. B.S. Dennis, V. Aksyuk, M.I. Haftel, S.T. Koev, G. Blumberg, Journ. Appl. Phys. 110, 066102 (2011). 11. S.T. Koev, A. Agrawal, H.J. Lezec, and V. Aksyuk, online reference: Plasmonics: DOI: 10.1007/S11468-011-9303. 12. T.W. Ebbesen, C. Genet, and S.I. Bozhevolnyi, Physics Today, May 2008, pp.44-50. 13. B. Lamprecht et al., Appl. Phys. Lett. 79, 51 (2001); P. Berini, Phys. Rev. B61, 10484 (2000); Phys. Rev. B63, 125417 (2001).